So the hairstyles of the Anglo-Saxons are not particularly well recorded, but neither are the hairstyles of most other European peoples of the day either. There are broadly speaking two ways to examine hairstyles in Medieval Europe, what did contemporary people say they wore, and what did they depict them wearing. This would be literary sources and artistic representations, but sadly neither is totally accurate and 100% the answer that we can use for everything.
So let's look first at the artistic depictions of various Anglo-Saxon rulers that were roughly contemporary. (We will deal with men first and look at women later on)
Here are two examples from the Bayeux Tapestry, which depicts the events of the Norman Conquest of England in 1066, it was commissioned and created a few decades later but was not a massively later creation. In these panels, the English kings Harold Godwinson and Edward the Confessor are shown with facial hair, a thin mustache in Harold's case and full beard in Edward's. If we go a generation prior, we can see that other Kings of England, if not Anglo-Saxons, such as Canute the Great were likewise depicted with beards, see here.
Here is Aethlstan for another depiction of a royal Anglo-Saxon figure, this being an extremely rare contemporary manuscript depiction.
So from these depictions, we may conclude that it was normal for Anglo-Saxon lords at the least to grow out their facial hair, but history is rarely so simple.
There are other contemporary depictions of these same figures, but without beards. On the coinage of Anglo-Saxon kings, they are almost never depicted with beards, or the "beards" are uncertain and may be a sign of wear on the coin and not a realistic artistic depiction.
Here are some Anglo-Saxon coins
So this changes things right? The depictions of them without beards are just as valid as those with them no? Well not exactly...
Coinage was not like currency today where the depictions of the monarchs, presidents, or whoever on the currency is a reasonably close likeness to the individual in question. Rather, the depiction of medieval monarchs on the coinage of the day was subject to a number of artistic and ideological considerations. Most Anglo-Saxon coins actively mimicked the coins of Arabic, Frankish, and above all Roman antecedents, and Roman coinage rarely depicted the real form of emperors, with beards and the like. Rather, they cut an idealized image of the ruler, and the same was true of early Medieval coins as well.
So between these two forms of evidence, I am inclined to give more credence to the non-numismatic evidence, and operate off of the assumption that facial hair, both mustaches and beards, was in style for Anglo-Saxon lords. From written sources we know that especially post-Viking Age, the hairstyles of men in England tended to the long, extravagant, and well maintained for both facial hair and for their head of hair. (As early as the late 8th century it seems that fashion of England for lords moved towards long beards and long hair, in imitation of their Scandinavian adversaries.
However not all men likely wore their hair in such styles, as seen on the Bayeux tapestry and in some other contexts, it looks as though many of the men are beardless and wear their hair shorter than the lords. This indicates a gap in the acceptability of wearing long hair on men for slaves, servants, and other members of society in lower social station.
What about the women?
This is sadly much harder to parse out for two reasons.
One, there are few contemporary depictions of women from this time. This applies across the board to women who were of royal or noble status as well as lower class women. There are simply very few depictions of them that are contemporary to our time period, the 8-11th centuries.
A few that I have found are:
Aethelflaed of Wessex Though this depictions is 13th century and she lived in the 9th century.
And here is again Canute and his wife
Now if you look closely you may notice a common theme. Among the contemporary depictions of Anglo-Saxon elite women, their hairs is covered by a wimple. This make it rather difficult to fully describe the hair styles of elite women in Early English society, as quite simply the surviving depictions show them dressed in a style that hides their hair. This was actually quite common in the Middle Ages as a whole as women wearing their hair down, particularly early on, had certain connotations. Connotations that no respectable English royal or noble woman would want reflected on her.
Sadly our literary sources are equally silent. Beyond an apocryphal story about the Lady Godifu and her hair that was long enough to preserve her modesty on her journey through the streets of England, there are not many literary descriptions of women's hair in Anglo-Saxon England. it is tempting to instead fill in the gap of our knowledge with Scandinavian fashions as the men were known to adopt, but we lack the clear cut evidence that we have for male adoption of Scandinavian hairstyles.
In addition to the answer you already got about early medieval England, you might be interested in a previous answer of mine about hairstyles in early medieval Scotland.
